What is the value of $\sin(x)$ for the maximum value of $(5+3\sin(x))^2 (7-3\sin(x))^3$.
closed as off-topic by choco_addicted, Brahadeesh, José Carlos Santos, Claude Leibovici, Delta-u Oct 17 at 9:25
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AM-GM inequality is the neat approach, but if you can't figure it out algebra also helps...
let $z=\sin(x)$, $f(z)=g(z)^2h(z)^3$
$$ f' = 2gg'h^3 + 3gh^2h' = gh^2(2g'h+3hh') $$
since $-1 \le z \le 1$, $g$ and $h$ do not have zeros in this region. Therefore $2g'h=-3hh'$, which will give you the same equation to solve for $z$.